\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey

  • * Corresponding author: Mingxin Wang

    * Corresponding author: Mingxin Wang
This work was supported by NSFC Grant 11371113.
Abstract Full Text(HTML) Related Papers Cited by
  • In this paper, a diffusive prey-predator model with strong Allee effect growth rate and a protection zone $\Omega _0$ for the prey is investigated. We analyze the global existence, long time behaviors of positive solutions and the local stabilities of semi-trivial solutions. Moreover, the conditions of the occurrence and avoidance of overexploitation phenomenon are obtained. Furthermore, we demonstrate that the existence and stability of non-constant steady state solutions branching from constant semi-trivial solutions by using bifurcation theory. Our results show that the protection zone is effective when Allee threshold is small and the protection zone is large.

    Mathematics Subject Classification: Primary: 92D25, 92D50; Secondary: 35B32, 35B35, 35B40, 35K57.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] W. C. Allee, Principles of Animal Ecology, Saunders, RI, 1949.
    [2] H. R. AkcakayaR. Arditi and L. R. Ginzburg, Ratio-dependent prediction: An abstraction that works, Ecology, 76 (1995), 995-1004. 
    [3] R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73 (1992), 1544-1551.  doi: 10.2307/1940007.
    [4] R. G. Casten and C. G. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Diff. Equat., 27 (1978), 266-273.  doi: 10.1016/0022-0396(78)90033-5.
    [5] R. H. CuiJ. P. Shi and B. Y. Wu, Strong Allee effect in a diffusive predator-prey system with a protection zone, J. Diff. Equat., 256 (2014), 108-129.  doi: 10.1016/j.jde.2013.08.015.
    [6] Y. H. Du and X. Liang, A diffusive competition model with a protection zone, J. Diff. Equat., 244 (2008), 61-86.  doi: 10.1016/j.jde.2007.10.005.
    [7] Y. H. DuR. Peng and M. X. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Diff. Equat., 246 (2009), 3932-3956.  doi: 10.1016/j.jde.2008.11.007.
    [8] Y. H. Du and J. P. Shi, A diffusive predator-prey model with a protection zone, J. Diff. Equat., 229 (2006), 63-91.  doi: 10.1016/j.jde.2006.01.013.
    [9] S. B. Hsu, On global stability of a predator-prey system, Math. Biosci., 39 (1978), 1-10.  doi: 10.1016/0025-5564(78)90025-1.
    [10] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025.
    [11] C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Ent., 91 (1959), 385-398.  doi: 10.4039/Ent91385-7.
    [12] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-New York, 1966.
    [13] P. E. Kloeden and C. Pötzsche, Dynamics of modified predator-prey models, Int. J. Bifurc. Chaos, 20 (2010), 2657-2669.  doi: 10.1142/S0218127410027271.
    [14] Y. Lou, Some reaction diffusion models in spatial ecology, Sci. Sin. Math., 45 (2015), 1619-1634.  doi: 10.1360/N012015-00233.
    [15] Y. Lou and B. Wang, Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment, J. Fixed Point Theory Appl., 19 (2017), 755-772.  doi: 10.1007/s11784-016-0372-2.
    [16] Y. Li and M. X. Wang, Stationary pattern of a diffusive prey-predator model with trophic intersections of three levels, Nonlinear Anal. RWA, 14 (2013), 1806-1816.  doi: 10.1016/j.nonrwa.2012.11.012.
    [17] R. M. May, Stability and Complexity in Model Ecosystems, Princeton Univ. Press, Princeton, 1973.
    [18] K. MischaikowH. Smith and H. R. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685.  doi: 10.1090/S0002-9947-1995-1290727-7.
    [19] N. Min and M. X. Wang, Qualitative analysis for a diffusive predator-prey model with a transmissible disease in the prey population, Comput. Math. Appl., 72 (2016), 1670-1689.  doi: 10.1016/j.camwa.2016.07.028.
    [20] L. Nirenberg, Topics in Nonlinear Functional Analysis, Amer. Math. Soc., Providence, RI, 2001. doi: 10.1090/cln/006.
    [21] W.J. Ni and M. X. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Diff. Equat., 261 (2016), 4244-4272.  doi: 10.1016/j.jde.2016.06.022.
    [22] W. J. Ni and M. X. Wang, Dynamicl properties of a Leslie-Gower prey-predator model with strong Allee effect in prey, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3409-3420.  doi: 10.3934/dcdsb.2017172.
    [23] P. Y. H. Pang and M. X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 919-942.  doi: 10.1017/S0308210500002742.
    [24] F. Rao and Y. Kang, The complex dynamics of a diffusive prey-predator model with an Allee effect in prey, Ecol. Complex., 28 (2016), 123-144.  doi: 10.1016/j.ecocom.2016.07.001.
    [25] M. X. Wang, Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion, Phys. D, 196 (2004), 172-192.  doi: 10.1016/j.physd.2004.05.007.
    [26] M. X. Wang, Nonlinear Elliptic Ppartial Differential Equations (in Chinese), Science Press, Beijing, 2010.
    [27] Y. X. Wang and W. T. Li, Effect of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone, Nonlinear Anal. RWA, 14 (2013), 224-245.  doi: 10.1016/j.nonrwa.2012.06.001.
    [28] J. F. WangJ. P. Shi and J. J. Wei, Dynamics and pattern formation in a diffusive predator-prey systems with strong Allee effect in prey, J. Diff. Equat., 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.
    [29] Q. X. Ye, Z. Y. Li, M. X. Wang and Y. P. Wu, The Introduction of Reaction-Diffusion Equations (in Chinese), Science Press, Beijing, 2011.
    [30] F. Q. YiJ. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Diff. Equat., 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.
    [31] J. Zhou and C. L. Mu, Coexistence of a diffusive predator-prey model with Holling type-Ⅱ functional response and density dependent mortality, J. Math. Anal. Appl., 385 (2012), 913-927.  doi: 10.1016/j.jmaa.2011.07.027.
  • 加载中
SHARE

Article Metrics

HTML views(840) PDF downloads(434) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return